There doesn't seem to be many. On Wikipedia's listing of cocktails, the only Q is the Quentão, which means "very hot. It's a warm Brazilian cocktail mixing cachaca and spices. It's definitely a regional drink, as it's hard to even acquire cachaca outside of Brazil.
On the cocktail database Drinksmixer, this is where all of the Q cocktails are:
http://www.drinksmixer.com/cat/298/92/
Lots of things starting with Queen or Quick, but nothing that'd I'd call well-known.
quaff = to drink deeply
quesadilla
which African American has a last name that starts with Q? which African American has a last name that starts with Q?
There have been no astronauts who's last name starts with the letter "Q".
Quinton
Suzi Quatro is a singer. Her name begins with the letter Q.
queen
quentin
quartz
· Quail
Quinn is a girl name. It begins with the letter Q.
Quiche, quail, quesadillas and Quaker Life cereal are foods. Qibla Cola (Qibla Cola Company), Qoo soft drink (Coca-Cola Company), Quadro vodka, Quail Springs IPA beer, Quaker's cocktail, Quebec cocktail and Quentao (a Brazilian cocktail drink) are drinks.